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Test for an Arithmetic Progression

T₂ - T₁ = T₃ - T₂ = d

Teat for a Geometric Progression

T₂ ÷ T₁ = T₃ ÷ T₂ = r

The term of an Arithmetic progression

Tn = a + (n-1) d

Tn = arⁿ⁻¹

The term of an Geometric progression

Sn = ½n( a+ l)

The sum of an Arithmetic progression if you know the last term

Sn = ½n( 2a+(n-1) d)

The sum of an Arithmetic progression if you DON'T know the last term

Sn = a(1 - rⁿ)/1 - r

The sum of a Geometric progression if r<1

Sn = a(rⁿ - 1)/ r- 1

The sum of a Geometric progression if r>1

Sum to Infinity

Only a geometric progression

Compound Interest

A = P(1+r/n)^(nt)

u1

first term of a sequence

n

number of terms in a sequence

l

last term of a sequence

d

common difference

R

common ratio

un

value of the nth term

Sn

sum of the first n terms

Sinfinity

sum to infinity

Arithmetic sequence

a sequence of numbers that has a common difference between every two consecutive terms

arithmetic sequence: recursive formula

find term from previous term

arithmetic sequence: general formula

find any term using n

geometric sequence

a sequence of numbers that has a common ratio between every two consecutive terms

- find general formula

when asked to find when a certain value reached in a sequence

Series

sum of the terms of a sequence

Arithmetic series

Sum of the terms of a sequence

geometric series

sum of the terms of an arithmetic sequence

divergent series

a series that approaches infinity

convergent series

a series that is approaching a limit

find limit

u1 / (1-r)

limit only limit if

can be approached from both sides

= L

notation of limit

graph function

find limit with GDC

Sequence

A set of numbers arranged in a special order or pattern.

pn(x)=1-((x^2)/2!)+((x^4)/4!)-((x^6)/6!)+....((x^n)/n!)

Maclauren series of cos(x)

divergent

harmonic series 1/N

MacLauren series of [1/(1-x)]

1 + x + x^2 + x^3 + x^4 ...+ x^n

1+x+(x^2)/(2!)+.....(x^n)/(n!)

MacLauren series of e^x

x-((x^3)/3!)+((x^5)/5!)-((x^7)/7!)+...

Maclaurin series of sin x

Alternating Harmonic Series

The series [ ( (-1) ^ n+1 ) / n ] ... converges from 1 to infinity

MacLauren series of ln(1 + x)

x - (x²/2) + (x³/3) - (x⁴/4) + .... + ((-1ⁿ) * (xⁿ/ n))

Nth Term Test

If Limit as K approaches infinity, then the Series of K Diverges.

P-Series

Sum where 1/(n^p) converges if and only if p > 1

Geometric Series Convergence Test

If IrI<1, then the geometric series Σar^n converges.

Integral Test

take integral and evaluate, if it goes to a number it is convergent, if there is ∞in it, its divergent

Alternating Series Test

If An is positive, the series ∑(-1)An converges if & only if..

Comparison Test

after comparing remember to take the limit as n→∞ of an/bn and for convergence it must lie on the interval 0<L<∞

1-(x^2)+(x^4)- ... +(-1)ⁿ((xⁿ)/n)

Maclaurin Series of ln(1+x^2)

Ratio Test

limit as n→∞ of an+1/an

limit comparison test

Let A and B be series with positive terms such that p = limit A/B (to infinity) and 0 < p < infinity, then both series converge or both diverge

Lagrange Error Bound

Error = M/((n+1)!) * (x-a)^(n+1) where M is the highest value on the interval

nth -degree Taylor polynomial

Pn(x) = ∑ (fk(a) / k! ) (x-a)^k

Maclaurin Series of 1/x-1

1 + x + x^2 + x^3....(xⁿ)

Alternating Series Error Bound

If Σa sub n is an alternating series in which |a sub n+1| < |a sub n| for all k, then the error in the nth partial sum is no larger than |a sub n+1|.

Absolute Convergence

∑An is convergent as well as ∑|An|

interval of convergence

set of x values for which power series converges

Taylor Series

If a function is differentiable infinitely many times in some interval around x=a, then the Taylor Series centered at a for f is:

Power Series

do the ratio test, then find the interval of convergence and where the origin is

1-x+x^2-x^3...(-1^n)(x^n)

McClaurin series for 1/(x+1)

x-(x^3/3)+(x^5)/5-....((-1)^x)*(x^(n+1))/(2n+1)

McClaurin series for acrtan(x)

Conditional Convergence

∑An is convergent but ∑|An| diverges

a=r^k

Geometric Sequence

Reciprocal of Factors

∑(1/n!) converges to e

arithmetic recursive

an = an-1 + d

arithmetic explicit

an = a₁ + (n-1)d

Sn = ½n(a₁+an)

arithmetic sum of the series

arithmetic sum of series

Sn = ½n(2a₁ + (n-1)d)

geometric recursive

gn = (gn-1)r

Geometric Explicit

a(n)=a x r to the n - 1

Sn = (g₁(1-r^n))/1 - r

sum of a finite geometric series

Sn = (g₁(r^n - 1))/r - 1

sum of a finite geometric series when r>1

Sn = g₁/(1-r)

sum of an infinite geometric series